Finite Group DatabasebetaFinite Group DatabasebetaOrder 31200.
115 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72224461 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | PΣL(2,𝔽25) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | PGL(2,𝔽25) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | PSL(2,𝔽25) | C22 | 1 | 1 | yes | characteristic |
| 5 | AΓL(1,𝔽25) | — | 26 | 1 | no | maximal |
| 6 | #64480⋊C2 | — | 26 | 1 | no | — |
| 7 | AGL(1,𝔽25) | — | 26 | 1 | no | — |
| 8 | #10597167 | — | 26 | 1 | no | — |
| 9 | #6053⋊C2 | — | 78 | 1 | no | — |
| 10 | #3874⋊C2 | — | 26 | 1 | no | — |
| 11 | #1043⋊C3 | — | 26 | 1 | no | — |
| 12 | #64480 | — | 26 | 1 | no | — |
| 13 | C2×S5 | — | 130 | 1 | no | — |
| 14 | D5⋊AGL(1,𝔽5) | — | 78 | 1 | no | — |
| 15 | #6053 | — | 26 | 1 | no | — |
| 16 | #6053 | — | 78 | 1 | no | — |
| 17 | (C5⋊D5)⋊C3 | — | 26 | 1 | no | — |
| 18 | #3874 | — | 52 | 1 | no | — |
| 19 | C2×A5 | — | 130 | 1 | no | — |
| 20 | S5 | — | 130 | 1 | no | — |
| 21 | S5 | — | 130 | 1 | no | — |
| 22 | C2×#261 | — | 300 | 1 | no | maximal |
| 23 | D52 | — | 78 | 1 | no | — |
| 24 | #1044 | — | 78 | 1 | no | — |
| 25 | #1043 | — | 26 | 1 | no | — |
| 26 | #914 | — | 325 | 1 | no | maximal, Hall |
| 27 | C52⋊C3 | — | 26 | 1 | no | Hall |
| 28 | A5 | — | 130 | 1 | no | — |
| 29 | C2×D13 | — | 300 | 1 | no | — |
| 30 | #261 | — | 300 | 1 | no | — |
| 31 | #261 | — | 300 | 1 | no | — |
| 32 | C5⋊D5 | — | 26 | 1 | no | — |
| 33 | C5×D5 | — | 156 | 1 | no | — |
| 34 | C2×S4 | — | 650 | 1 | no | — |
| 35 | #239 | — | 325 | 1 | no | — |
| 36 | D3×D4 | — | 325 | 1 | no | — |
| 37 | C3×SD16 | — | 325 | 1 | no | — |
| 38 | #215 | — | 325 | 1 | no | — |
| 39 | #213 | — | 325 | 1 | no | — |
| 40 | D24 | — | 325 | 1 | no | — |
| 41 | ΓL(1,𝔽25) | — | 325 | 1 | no | — |
| 42 | C2×AGL(1,𝔽5) | — | 780 | 1 | no | — |
| 43 | AGL(1,ℤ/8) | — | 975 | 1 | no | Sylow |
| 44 | C2×C13 | — | 300 | 1 | no | — |
| 45 | D13 | — | 300 | 1 | no | — |
| 46 | D13 | — | 300 | 1 | no | — |
| 47 | C52 | — | 26 | 1 | no | Sylow |
| 48 | C22×D3 | — | 650 | 1 | no | — |
| 49 | C2×A4 | — | 650 | 1 | no | — |
| 50 | S4 | — | 650 | 1 | no | — |
| 51 | S4 | — | 650 | 1 | no | — |
| 52 | C3×Q8 | — | 325 | 1 | no | — |
| 53 | C3×D4 | — | 325 | 1 | no | — |
| 54 | #67 | — | 650 | 1 | no | — |
| 55 | D12 | — | 325 | 1 | no | — |
| 56 | D12 | — | 325 | 1 | no | — |
| 57 | D12 | — | 650 | 1 | no | — |
| 58 | C4×D3 | — | 325 | 1 | no | — |
| 59 | C4×D3 | — | 650 | 1 | no | — |
| 60 | C3×C8 | — | 325 | 1 | no | — |
| 61 | C3⋊C8 | — | 325 | 1 | no | — |
| 62 | C2×D5 | — | 780 | 1 | no | — |
| 63 | AGL(1,𝔽5) | — | 780 | 1 | no | — |
| 64 | AGL(1,𝔽5) | — | 780 | 1 | no | — |
| 65 | #41 | — | 975 | 1 | no | — |
| 66 | C2×D4 | — | 975 | 1 | no | — |
| 67 | SD16 | — | 325 | 1 | no | — |
| 68 | SD16 | — | 975 | 1 | no | — |
| 69 | D8 | — | 975 | 1 | no | — |
| 70 | D8 | — | 975 | 1 | no | — |
| 71 | M16 | — | 975 | 1 | no | — |
| 72 | C13 | — | 300 | 1 | no | Sylow |
| 73 | C22×C3 | — | 650 | 1 | no | — |
| 74 | C2×D3 | — | 1300 | 1 | no | — |
| 75 | C2×D3 | — | 1300 | 1 | no | — |
| 76 | C2×D3 | — | 325 | 1 | no | — |
| 77 | C2×D3 | — | 650 | 1 | no | — |
| 78 | C2×D3 | — | 650 | 1 | no | — |
| 79 | A4 | — | 650 | 1 | no | — |
| 80 | C3×C4 | — | 325 | 1 | no | — |
| 81 | C3×C4 | — | 650 | 1 | no | — |
| 82 | Dic3 | — | 325 | 1 | no | — |
| 83 | C2×C5 | — | 780 | 1 | no | — |
| 84 | D5 | — | 780 | 1 | no | — |
| 85 | D5 | — | 156 | 1 | no | — |
| 86 | C23 | — | 650 | 1 | no | — |
| 87 | Q8 | — | 325 | 1 | no | — |
| 88 | D4 | — | 325 | 1 | no | — |
| 89 | D4 | — | 1950 | 1 | no | — |
| 90 | D4 | — | 1950 | 1 | no | — |
| 91 | D4 | — | 975 | 1 | no | — |
| 92 | D4 | — | 975 | 1 | no | — |
| 93 | C2×C4 | — | 1950 | 1 | no | — |
| 94 | C2×C4 | — | 975 | 1 | no | — |
| 95 | C8 | — | 325 | 1 | no | — |
| 96 | C8 | — | 975 | 1 | no | — |
| 97 | C2×C3 | — | 1300 | 1 | no | — |
| 98 | C2×C3 | — | 325 | 1 | no | — |
| 99 | D3 | — | 1300 | 1 | no | — |
| 100 | D3 | — | 1300 | 1 | no | — |
| 101 | D3 | — | 650 | 1 | no | — |
| 102 | C5 | — | 156 | 1 | no | — |
| 103 | C22 | — | 650 | 1 | no | — |
| 104 | C22 | — | 650 | 1 | no | — |
| 105 | C22 | — | 1950 | 1 | no | — |
| 106 | C22 | — | 975 | 1 | no | — |
| 107 | C4 | — | 325 | 1 | no | — |
| 108 | C4 | — | 650 | 1 | no | — |
| 109 | C4 | — | 975 | 1 | no | — |
| 110 | C3 | — | 325 | 1 | no | Sylow |
| 111 | C2 | — | 300 | 1 | no | — |
| 112 | C2 | — | 325 | 1 | no | — |
| 113 | C2 | — | 130 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | PSL(2,𝔽25) | C22 |
| LCS | G ▷ PSL(2,𝔽25) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72224461 | C2 | semidirect |
| 2. PΣL(2,𝔽25) | C2 | semidirect |
| 3. PGL(2,𝔽25) | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72224461 | 2 |
| PΣL(2,𝔽25) | 2 |
| PGL(2,𝔽25) | 2 |
| AΓL(1,𝔽25) | 26 |
| C2×#261 | 300 |
| #914 | 325 |